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Properties

Label 2-2156-7.2-c1-0-23
Degree $2$
Conductor $2156$
Sign $-0.266 + 0.963i$
Analytic cond. $17.2157$
Root an. cond. $4.14918$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (1 − 1.73i)9-s + (−0.5 − 0.866i)11-s + 4·13-s + 0.999·15-s + (−3 − 5.19i)17-s + (−1 + 1.73i)19-s + (−0.5 + 0.866i)23-s + (2 + 3.46i)25-s − 5·27-s + 2·29-s + (−0.5 − 0.866i)31-s + (−0.499 + 0.866i)33-s + (4.5 − 7.79i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.223 + 0.387i)5-s + (0.333 − 0.577i)9-s + (−0.150 − 0.261i)11-s + 1.10·13-s + 0.258·15-s + (−0.727 − 1.26i)17-s + (−0.229 + 0.397i)19-s + (−0.104 + 0.180i)23-s + (0.400 + 0.692i)25-s − 0.962·27-s + 0.371·29-s + (−0.0898 − 0.155i)31-s + (−0.0870 + 0.150i)33-s + (0.739 − 1.28i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(17.2157\)
Root analytic conductor: \(4.14918\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2156} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2156,\ (\ :1/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.250465639\)
\(L(\frac12)\) \(\approx\) \(1.250465639\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.5 + 7.79i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5 + 8.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11T + 71T^{2} \)
73 \( 1 + (7 + 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-6.5 + 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.970223580286368908302998071960, −7.896262769461582017507628713383, −7.27154564983345459006577599381, −6.46861111383483806530564430342, −5.94904164635695306718978825712, −4.83754937906546296450857556127, −3.82746184643946894282068710752, −3.02603202034414322023880640952, −1.72519382814047571503695776360, −0.49293345069549535176435835483, 1.28439983828070620310397239280, 2.47893070609580533230786152337, 3.84991415018960788003360682028, 4.41758361444581985897074544700, 5.18838588539747291173815597652, 6.17791496282122700598820513981, 6.82616851982956168273645930075, 8.048262074180668912546398423150, 8.413600060032320351069710890438, 9.271203085496987075978915162261

Graph of the $Z$-function along the critical line